Young's Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

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JEE : Young's Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

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Young’s Modulus

For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

It has the unit of longitudinal stress and dimensions of [MI. -1T-2] Its unit is Pascal or N/m2.

Bulk Modulus

Within elastic limit the bulk modulus is defined as the ratio of longitudinal stress and volumetric strain. It is given as:
Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev
– ve indicates that the volume variation and pressure variation always negate each other.

  •  Reciprocal of bulk modulus is commonly referred to as the “compressibility”. It is defined as the fractional change in volume per unit change in pressure.
  • Shear Modulus or Modulus of Rigidity

It is defined as the ratio of the tangential stress to the shear strain.

Modulus of rigidity is given by
 

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

  •  Poisson’s Ratio

The ratio of change in diameter (ΔD) to the original diameter (D) is called lateral strain. The ratio of change in length (Δl) to the original length (l) is called longitudinal strain. The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

For most of the substances, the value of σ lies between 0.2 to 0.4 When a boay is perfectly incompressible, the value of σ is maximum and equals to 0.5.

  • Elastic Fatigue

It is the property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming forces.

  •  Relations between Elastic Moduli

For isotropic materials (i.e., materials having the same properties in all directions), only two of the three elastic constants are independent. For example, Young’s modulus can be expressed in terms of the bulk and shear moduli.
Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

  •  Breaking Stress

The ultimate tensile strength of a material is the stress required to break a wire or a rod by pulling on it. The breaking stress of the material is the maximum stress which a material can withstand. Beyond this point breakage occurs.

  • When a wire of original length L is stretched by a length 1 by the applied I inn of force F at one end, then

Work done to stretch wire  Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

  • Work done per unit volume of wire is given as:

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

According to the formula given by

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Where F is the force needed to stretch the wire of length L and area of cross-section A,  is the increase in the length of the wire.

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

The work done by this force in stretching the wire is stored in the wire as potential energy.

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Integrating both sides, we get

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev
Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Which equal to the elastic potential energy U.

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Now the potential energy per unit volume is

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev
Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Young`s Modulus, Bulk Modulus and Elastic Moduli Class 11 Notes | EduRev

Hence, the elastic potential energy of a wire (energy density) is equal to half the product of its stress and strain.

  •  IMPORTANT TABLES

TABLE 9.1 Young's modulus, elastic limit and tensile strengths of some materials.

Substance

Young's modulus 109N/m2

σy

Elastic limit 107N/m2

%

Tensile strength 107N/m2

σu

Aluminium

70

IS

20

Copper

120

20

40

Iron {wrought)

190

17

33

Steel

200

30

50

Bone

 

 

 

(Tensile)

16

 

12

(Compressive)

9

 

12

 

TABLE 9.2 Shear modulus (G) of some common materials

Material

G(109 Nm-2 or GPa)

Aluminium

25

Brass

36

Copper

42

Glass

23

Iron

70

Lead

5.6

Nickel

77

Steel

84

Tungsten

150

Wood

10

 

TABLE 9.3 Bulk modulus (B) of some common Materials

Material Solids

B(109 Nm-2 or GPa)

Aluminium

72

Brass

61

Copper

140

Glass

37

Iron

100

Nickel

260

Steel

160

Liquids

 

Water

2.2

Ethanol

0.9

Carbon disulphide

1.56

Glycerine

4.76

Mercury

25

Gases

 

Air (at 5TP)

1.0 * 10-4

 

TABLE 9.4 Stress, strain and various elastic modulus

 

Type of Stress

Stress

Strain

Change in

Elastic

modulus

Name of
 modulus

State of Mater

   shapevolume   

Tensile

or

Compressive

Two equal and opposite forces perpendicular to Opposite faces (σ = F/A)

Elongation or compression parallel to force direction (ΔL/L) (longitudinal strain)

Yes

No

V = (FxL) /(A » ΔL)

Young's

modulus

Solid

Shearing

Two equal and opposite forces parallel to opposite surfaces [forces in each case such that total force and total torque on the body vani-shes (σs * F/A)]

Pure shear, θ

Yes

No

C - (F x θ) /A

Shear

modulus

Solid

Hydraulic

Forces perpendicular every where to the surface, force per unit area (pressure) same everywhere

Volume change (compression or elongation) (ΔV/V)

No

Yes

B = - p/(Δ V)

Bulk

modulus

Solid,

liquid and gas

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